Recently, motivated by supersymmetric gauge theory, Cachazo, Douglas, Seiberg, and Witten proposed a conjecture about finite dimensional simple Lie algebras, and checked it in the classical cases. We prove the conjecture for type G2, and also verify a consequence of the conjecture in the general case. 1. The CDSW conjecture Let g be a simple finite dimensional Lie algebra over C. We fix an inva...

Recently, motivated by supersymmetric gauge theory, Cachazo, Douglas, Seiberg, and Witten proposed a conjecture about finite dimensional simple Lie algebras, and checked it in the classical cases [CDSW,W]. Later V. Kac and the author proposed a uniform approach to this conjecture [EK], based on the theory of abelian ideals in the Borel subalgebra; this allowed them to check the conjecture for t...

Real life graphs and networks are prone to failure of nodes (vertices) and links (edges). In particular, for a pair of nodes s and t and a failing edge e in a graph G, the replacement path πG−e(s, t) is a shortest s−t path that avoids e. In this paper we focus on the general case in which (s, t) ∈ S × T , where S, T ⊆ V (G), and we present several efficient constructions maintaining the collect...

In this paper, we construct a lagrangian subspace of the space of representations Mg,l := HomC(πg,l, U)/U of the fundamental group πg,l of a punctured Riemann surface into an arbitrary compact connected Lie group U . This lagrangian subspace is obtained as the fixed-point set of an anti-symplectic involution β̂ defined on Mg,l. We show that the involution β̂ is induced by a form-reversing involut...

The ability to directly follow and time-resolve the rearrangement of the nuclei within molecules is a frontier of science that requires atomic spatial and few-femtosecond temporal resolutions. While laser-induced electron diffraction can meet these requirements, it was recently concluded that molecules with particular orbital symmetries (such as πg) cannot be imaged using purely backscattering ...

The generalized prism πG of G is the graph consisting of two copies of G, with edges between the copies determined by a permutation π acting on the vertices of G. We define a generalized Cartesian product G π H that corresponds to the Cartesian product G H when π is the identity, and the generalized prism when H is the graph K2. Burger, Mynhardt and Weakley [On the domination number of prisms o...

Power moments of Jk(t, G) = 1 √ πG ∫ ∞ −∞ |ζ( 1 2 + it+ iu)|2ke−(u/G)2 du (t ≍ T, T ε ≤ G ≪ T ), where k is a natural number, are investigated. The results that are obtained are used to show how bounds for ∫ T 0 |ζ( 1 2 + it)|2k dt may be obtained.

Goal: Describe the image of the cone Sn ≥� of positive semidefinite quadratic forms under the projection πG . Sn ≥�: convex cone of quadratic forms∑i , j ai jxix j such that∑i , j ai jpi p j ≥ � for all (p�, . . . , pn) ∈ Rn. Theorem (Diagonalization ofQuadratic Forms). Aquadratic form q ∈ R[x�, . . . , xn] is positive semidefinite if and only if it is a sum of squares of linear forms after a c...

At the end of Section 2 of the main paper, the semiparametric model framework within which we derive the proposed methods is stated, and is repeated here for convenience. The data from a clinical trial are denoted by (Yi, Xi, Zi), i = 1, . . . , n, assumed iid across i, where Y denotes the response of interest; X is a vector of baseline auxiliary covariates; and Z = 1, . . . , k depending on to...

An identifying code in a graph is a dominating set that also has the property that the closed neighborhood of each vertex in the graph has a distinct intersection with the set. The minimum cardinality of an identifying code in a graph G is denoted γ(G). We consider identifying codes of the direct product Kn × Km. In particular, we answer a question of Klavžar and show the exact value of γ(Kn×Km...